On Nietzsche... From Will To Power:
"At the same time I grasped that my instinct went into the opposite direction from Schopenhauer's: toward a justification of life, even at its most terrible, ambiguous, and mendacious; for this I had the formula Dionysian. Against the theory that an "in-itself of things" must necessarily be good, blessed, true, and one, Schopenhauer's interpretation of the "in-itself" as will was an essential step; but he did not understand how to deify this will: he remained entangled in the moral-Christian ideal. Schopenhauer was still so much subject to the dominion of Christian values that, as soon as the thing-in-itself was no longer "God" for him, he had to see it as bad, stupid, and absolutely reprehensible. He failed to grasp that there can be an infinite variety of ways of being different, even of being god."
I'm not well-versed in Nietzsche, but one thing I've retained from my reading of him is that Nietzsche thought the "ascetic" pessimism of Schopenhauer and his acolytes was detrimental to the flourishing of "great" people. Nietzsche did not reject pessimism but he tried to find a different way of approaching it in a way that ultimately affirmed life, because there are things in life that are beautiful, sublime, etc. At the core of his thought seems to be this notion of "health" - that no matter the circumstances the "healthy" person is able to flourish, and that the ascetics were really simply sick and diseased.
So Nietzsche was concerned that the influence of Schopenhauer's pessimism on the continent was negatively impacting the lives of people who would otherwise go on and do great things. This of course includes the production of music which Nietzsche criticized (like Wagner et al). It seems as though Nietzsche thought reading Schopenhauer dissolved potential in people. Nietzsche seemed to have wanted to instill a new sense of purpose and meaning in people so this wouldn't keep happening.
Nietzsche's philosophy was a product of the current cultural shift happening in the continent at the time. He's important, sure, but he is studied too much and given too much credit for ideas that weren't even his per se. It wasn't just Schopenhauer ---> Nietzsche, it was Schopenhauer, Feuerbach, Frauenstadt, Duhring, von Hartmann, Mainlander, Bahnsen, etc etc.
It might be the case that Nietzsche is so wildly popular simply because ascetic pessimism is not altogether that satisfactory. Sooner or later people get bored and want more and it's refreshing to hear someone speak about active power and drama and achievement and heroism and all that.